Then provide the keys to those who you would like to avoid using passwords.

ssh-keygen is used to generate that key pair for you. Here is a session where your own personal private/public key pair is created:

#ssh-keygen -t rsa

The command ssh-keygen -t rsa initiated the creation of the key pair.

I didn't enter a passphrase for my setup (Enter key was pressed instead).

The private key was saved in .ssh/id_rsa. This file is read-only and only for you. No one else must see the content of that file, as it is used to decrypt all correspondence encrypted with the public key.

The public key is save in .ssh/id_rsa.pub.

Its content is then copied in file .ssh/authorized_keys of the system you wish to SSH to without being prompted for a password.